At a news conference following the talks, Mr Trump said: “The relationship between our two nations is indispensable to the cause of liberty, justice, and peace.”

He also said:

He had apologised to Mrs May for the Sun’s story – but she told him not to worry as “it’s only the press”

He said had not given Mrs May advice on how to deal with the EU but “did give her a suggestion… and I think she found it maybe too brutal”

Brexit was an “incredible opportunity” and “whatever” the UK did after it left the EU was “OK with me”

Mr Trump described Brexit as a “very tough situation… between the borders and the entries into the countries and all of the things”, saying: “The only thing I ask is that she work it out so that we can have very even trade.”

Mrs May said the US was “keen” to do a deal with the UK, adding: “We will do a trade deal with them and with others around the rest of the world.”

She maintained the government’s Brexit agreement “delivers” on the referendum vote.

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Earlier, Mr Trump said he and Mrs May had spoken for an hour-and-a-half at the black-tie dinner at Blenheim Palace on Thursday, which he attended with his wife Melania.

“I think we probably never developed a better relationship than last night,” he said.

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Mr Trump arrived at the prime minister’s Buckinghamshire residence by helicopter after visiting the Royal Military Academy at Sandhurst, Berkshire, on Friday morning.

Meanwhile, Mrs Trump played bowls with the PM’s husband, Philip May, at the Royal Hospital Chelsea in London. She met Chelsea Pensioners and local children.

Downing Street said Mrs May had presented the US president with a gift of an illustrated ancestral chart of his Scottish heritage through his mother, and his wife with a bespoke perfume called the First Lady in a custom bottle.

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In his Sun interview,Mr Trump also said former Foreign Secretary Boris Johnson – who disagrees with the PM on Brexit and resigned this week – would make a “great prime minister”, adding: “I think he’s got what it takes.”

At the Chequers news conference, Mr Trump said he had been responding to the Sun’s question about Mr Johnson as a possible prime minister, adding: “He has been very nice to me. He’s been saying very good things about me as president”.